Representations of cartesian product $G$ We know for both representations of a locally compact group $G$, their tensor product is a representation of $G \times G$ (cartesian product of $G$ with self). Is each representation of it of this form?
 A: This is false, even when $G$ is finite but it's not far off from being true. If $G$  and $H$ are finite groups, irreducible representations of $G \times H$ are of the form $\{V \otimes W\}$ with $V, W$ irreducible representations of $G, H$ respectively. So, representations of $G \times H$ are direct sums of tensor products of representations of $G$ and $H$. But if $V_{1}, V_{2}$ and $W_{1}, W_{2}$ are nonisomorphic representations of $G$ and $H$ respectively, then
$$V_{1} \otimes W_{1} \oplus V_{2} \otimes W_{2}$$
cannot be written as $V \otimes W$ for any $V, W$ representations of $G, H$.
In general, for two locally compact groups $G, H$, the continuous finite dimensional representations of $G \times H$ are direct sums of tensor products of continuous finite dimensional representations of $G$ and $H$. This can be restated by saying
$$\mathrm{Rep}_{\mathbb{C}}(G \times H) = \mathrm{Rep}_{\mathbb{C}}(G) \boxtimes \mathrm{Rep}_{\mathbb{C}}(H)$$
where the tensor product on categories is known as Deligne tensor product or external tensor product. The objects in $\mathcal{C} \boxtimes \mathcal{D}$ are direct sums of tensor products of objects in $\mathcal{C}, \mathcal{D}$.
I believe the same happens for infinite dimensional representations too but I'm not 100% certain.
